Calculate the derivative using Part 2 of the Fundamental Theorem of Calculus

Question

Enclose arguments of functions in parentheses. For example, cos(2x)

Answer 1

d/dx(∫_x8^{e^3x}ln(t)dt) = ln(e^3x)d/dx(e^3x) - ln(x⁸)d/dx(x⁸) = 9xe^3x - 64x⁷lnx

Answer 2

∫_{(from x^8 to 1)} lnt dt + ∫_{(from 1 to e^3x)} lntdt = ∫_{(from 1 to e^3x)}lnt dt - ∫_{(from 1 to x^8)}lntdt

Using the FTC to take the derivative, we get ln(e^3x)(e^3x)' - ln(x⁸)(x⁸)'

= 3x(3e^3x) - 8lnx(8x⁷) = 9xe^3x - 64x⁷lnx